Displacement Modifier and Texture Coordinates
Local is when you are using the object's own XYZ coordinates, regardless of it's placement in the global coordinates. For example, When you rotate an object in object mode the object will no longer be aligned with the world



Global has to do with the XYZ grid that starts in the origin 0,0,0. The transformations occur using the world's coordinates.

Here's a small example of how If your object is not aligned with the global coordinates you'll get different results using the displace modifier: